President Obama and House Speaker John A. Boehner moved close to agreement Monday on a plan to avert the year-end “fiscal cliff,” but they had yet to clear several critical hurdles, including winning the support of wary House Republicans.

Obama and Boehner (R-Ohio) huddled at the White House for 45 minutes Monday morning for their third conversation in the past five days. Later, Boehner met for an hour at the Capitol with his leadership team in advance of a briefing Tuesday morning for the entire House GOP that could be a crucial test of Boehner’s ability to sell the deal.

Behind the scenes, administration officials and senior Republican aides continued to make progress. Obama laid out a counteroffer that included significant concessions on taxes, reducing the amount of new revenue he is seeking to $1.2 trillion over the next decade and limiting the hike in tax rates to households earning more than $400,000 a year. Obama had previously sought $1.4 trillion in new revenue, with tax increases on income over $250,000.

Obama also gave ground on a key Republican demand — applying a less-generous measure of inflation across the federal government. That change would save about $225 billion over the next decade, with more than half the savings coming from smaller cost-of-living increases for Social Security beneficiaries.

In addition, Obama increased his overall offer on spending cuts and dropped his demand for extending the payroll tax holiday, which has benefited virtually every worker for the past two years. But he is still seeking $80 billion in new spending on infrastructure and unemployment benefits and an increase in the federal government’s borrowing limit large enough to avert any new fight over the issue for two years.

Boehner has offered a one-year debt-limit increase, and the fresh stimulus spending remains a sticking point, according to senior Republican aides, who also complained that the overall deal remains too tilted toward new taxes.

“Any movement away from the unrealistic offers the President has made previously is a step in the right direction, but a proposal that includes $1.3 trillion in revenue for only $930 billion in spending cuts cannot be considered balanced,” Boehner spokesman Michael A. Steel said in a written statement.

“We hope to continue discussions with the President so we can reach an agreement that is truly balanced and begins to solve our spending problem,” he said.

Talks over the fiscal cliff have accelerated since Boehner made an offer Friday to raise tax rates on income over $1 million and to delay a fight over the government’s borrowing limit in exchange for significant cuts to health and retirement programs.

On Monday, it became clear that the two sides are extremely close on the broad outlines of the deal that has eluded them for much of the past two years. Boehner’s latest offer calls for $2 trillion in savings over the next decade, half from higher taxes and half from cuts to the fast-growing health and retirement programs that are the federal government’s largest expense. All told, Obama’s latest offer calls for about $2.15 trillion in savings.

People in both parties said the next few days could prove critical: Either Obama and Boehner will reach a consensus and sell it to their respective parties, or talks will again collapse, leaving congressional leaders scrambling for a fallback plan to mitigate the economic damage from more than $500 billion in automatic tax increases and spending cuts scheduled to begin immediately after Dec. 31.

Both sides cautioned that rank-and-file lawmakers had yet to see — much less digest — legislation that would require them to breach fundamental party orthodoxies. For Democrats, Social Security benefits have been sacred. Republicans, meanwhile, would have to not only raise taxes but also agree to raise rates on successful individuals they have dubbed “job creators.”

Of particular concern to negotiators is the conservative wing of Boehner’s caucus, which has been boisterous in its opposition to tax increases of any kind. Rep. Jason Chaffetz (R-Utah) said he found initial reports of Boehner’s tax offer upsetting.

“I don’t like it,” Chaffetz said. But echoing a number of his conservative colleagues, he said he is willing to “give the speaker the benefit of the doubt and hear him out before I pass judgment.”

Liberal Democrats, too, were on edge about Obama’s offer on the inflation measure, known as the chained consumer price index, or chained CPI. Obama tentatively embraced the change in budget negotiations with Boehner in the summer of 2011. Since then, Senate Majority Leader Harry M. Reid (D-Nev.) and other Democrats have rejected the proposal, insisting that the solvency of Social Security should be addressed separately from the year-end negotiations.

Obama is seeking adjustments to blunt the impact on the very old and the infirm. His offer proposes to exclude disability payments, known as Supplemental Security Income, or SSI, and to provide a bump-up in benefits for retirees who reach age 85, Democrats said.

Rep. Donna Edwards (D-Md.), who organized 57 House Democrats to sign a letter last week urging Obama and congressional leaders to protect Social Security, nonetheless argued that adopting chained CPI would result in “serious benefit cuts for recipients, particularly for our seniors and the disabled.”

Meanwhile, the possibility remains that the deal could get even more distasteful for Democrats, particularly if Republicans counter Obama’s request for $1.2 trillion in new taxes with a demand for an additional concession on health care, such raising as the eligibility age for Medicare beneficiaries from 65 to 67.

While Obama tentatively agreed to raise the Medicare eligibility age during talks with Boehner in 2011, it has emerged in the current negotiations as a line the White House is unwilling to cross.

If a deal is reached, people in both parties say, it would include postponement of roughly $100 billion in automatic spending cuts to the budgets of the Pentagon and other agencies next year. But $1 trillion in future cuts to agency budgets adopted during the 2011 budget battle would remain in place. Counting around $800 billion in savings from ending the wars in Iraq and Afghanistan, the overall package would come close to reducing deficits by a total of $4 trillion over the next decade — a target that economists say would stabilize the soaring federal debt.

As talks continued, Reid made tentative plans to advance fiscal-cliff legislation in the Senate, warning lawmakers that they probably do not have time to finish before Christmas. Instead, Reid said the Senate would probably have to return to Washington next week to take a final vote.

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